Activity: Using Implicit Differentiation to Find Partial Derivatives of Equilibrium Variables

Generality of Directional Effects of Demand Shocks on Market Equilibrium

Analyzing a Market Shock with Calculus

Consider a competitive market where the quantity demanded is given by the function Q_d = a - bP and the quantity supplied is given by Q_s = c + dP. The variables P and Q represent price and quantity, respectively. The terms a, b, c, and d are all positive parameters that determine the positions and slopes of the curves. First, find the equilibrium price (P*) as a function of these parameters. Then, by taking the partial derivative of the equilibrium price with respect to the demand parameter 'a', identify the expression that correctly represents the rate of change of the equilibrium price as 'a' changes.

Economic Interpretation of a Partial Derivative in Market Analysis

Evaluating Methods for Comparative Statics

A necessary first step in using calculus for comparative statics analysis is to derive the explicit algebraic solution for the equilibrium price (P*) as a function of the model's parameters. Only after finding this explicit function can one take its partial derivative with respect to a parameter to determine the parameter's effect on the equilibrium price.

Consider a competitive market model where quantity demanded is Q = a - bP and quantity supplied is Q = c + dP. P is the price, Q is the quantity, and a, b, c, d are positive parameters. The parameter 'a' represents a demand-side factor (like consumer income), and 'c' represents a supply-side factor (like technology). P* and Q* represent the equilibrium price and quantity. Match each partial derivative with its correct economic interpretation.

An economist is analyzing a market model where supply and demand are represented by algebraic equations. To determine the direction of change in the equilibrium price when a single external factor (represented by a parameter) changes, a specific analytical procedure using calculus is followed. Arrange the steps of this procedure in the correct logical order.

Consider a competitive market where the quantity demanded is given by Q = 100 - 2P and the quantity supplied is given by Q = 10 + P + 0.5T. In this model, P is the price, Q is the quantity, and T is a parameter representing the level of production technology. To find the instantaneous rate of change of the equilibrium price with respect to a change in technology, you would first express the equilibrium price (P*) as a function of the parameter T and then compute the derivative. The value of this derivative, ∂P*/∂T, is ____.

Consider a competitive market where the quantity demanded is given by the function Q_d = Y/P and the quantity supplied is Q_s = P. In this model, P is the price, Q is the quantity, and Y is a parameter representing average consumer income. Using calculus, analyze how the equilibrium price (P*) responds to a change in consumer income (Y).

An economist models a competitive market and finds that for a specific parameter, 'γ' (gamma), the effect on the equilibrium price (P*) and quantity (Q*) is described by the following partial derivatives: ∂P*/∂γ < 0 and ∂Q*/∂γ > 0. Based on these mathematical results, which of the following real-world events could a change in the parameter 'γ' represent?